论文标题
$ c^0 $的清晰度与Lipschitzian扰动的非自主微分方程
Sharpness of $C^0$ conjugacy for the non-autonomous differential equations with Lipschitzian perturbation
论文作者
论文摘要
非自主微分方程的经典$ C^0 $线性化定理指出了非线性系统及其线性部分之间存在$ C^0 $拓扑结合。也就是说，存在同态（等效函数）$ h $，将非线性系统的解决方案发送到其线性部分的解决方案。在先前的文献中证明，如果非线性扰动是Lipschitzian，则等效函数$ h $及其反向$ g = h^{ - 1} $是连续的。问题：有可能提高规律性吗？规律性尖锐吗？为了回答这个问题，我们构建了反例，以表明等效函数$ h $完全是Lipschitzian，但是倒数$ g = h^{ - 1} $仅是Hölder的连续。此外，我们提出了一个猜想，即同态的这种规律性很敏锐（不再改善）。我们证明，猜想对于线性收缩的系统是正确的。此外，我们介绍了与频谱密切相关的线性扰动的特殊情况。
The classical $C^0$ linearization theorem for the non-autonomous differential equations states the existence of a $C^0$ topological conjugacy between the nonlinear system and its linear part. That is, there exists a homeomorphism (equivalent function) $H$ sending the solutions of the nonlinear system onto those of its linear part. It is proved in the previous literature that the equivalent function $H$ and its inverse $G=H^{-1}$ are both Hölder continuous if the nonlinear perturbation is Lipschitzian. Questions: is it possible to improve the regularity? Is the regularity sharp? To answer this question, we construct a counterexample to show that the equivalent function $H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely Hölder continuous. Furthermore, we propose a conjecture that such regularity of the homeomorphisms is sharp (it could not be improved anymore). We prove that the conjecture is true for the systems with linear contraction. Furthermore, we present the special cases of linear perturbation, which are closely related to the spectrum.
